I. Dichotomy (The Stadium)

To cross a unit space interval in a unit time, a runner must first cross half the interval in half the time, half of that half in one quarter of the time, and so forth over infinitely many points ordered in the sequence 1/2, 1/4, 1/8 .... But it is impossible to touch infinitely many points in a finite time, so it is impossible for the runner to reach his goal.